The mortar element method for three dimensional finite elements

M2AN - M

F. B EN B ELGACEM

Y. M ^ADAY

The mortar element method for three dimensional finite elements

M2AN - Modélisation mathématique et analyse numérique, tome 31, n^o2 (1997), p. 289-302

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MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE

iVol 31, n° 2, 1997, p 289 a 302)

THE MORTAR ELEMENT METHOD FOR THREE DIMENSIONAL FINITE ELEMENTS (*)

by F BEN BELGACEM ( 0 and Y MADAY (2)

Abstract — We present m thts paper the application of the mortar element method m the case where fini te element methods are used in each subdornains The novelty is m the posstbility of dealing with tenahedra for thefimte element mesh Thts paper thus complètes the analysis of the mot tai element methods that allows for couphng different spectral and/or finite element methods in adjascent mbdomains in a flexible and optimal way

Résume —Dans cet article nous presentons la methode des éléments avec joints appliquée au cas ou des éléments finis sont utilises sut chaque sous domaine La nouveauté teside dans la possibilité de considérer en 3D des maillages tetrêdnques Cet article complete donc l analyse de la methode des éléments avec joints qui permet un couplage flexible et optimal de diffetentes disetetisations spectiale éléments finis sut des sous domaines adjacents

1. INTRODUCTION

When domain décomposition is used for the approximation of the solution of some partial differential équation, a large problem is sphtted up mto a set of smaller ones that can, for example, fit easily on each processor of a parallel machine This ability can be used further on by tuning the approximation technique to the proper caractenstic of each smaller problem In this direction, ît cornes naturally mto mind that one could use, locally to each subdomam, the proper discretization parameter, and maybe even, the proper discretization, adapted to the local behaviour of the solution The mortar element method was mvented origmally (in 1987) to provide an optimal tooi in this framework

We refer to [9] and [10] for a gênerai présentation of the mortar element method, to [12] and [4] for PhD thesis on the coupbng of finite element method with spectral element methods m 2D and 3D respectively, to [16] and [2] (see also [3]) for the couphng m the pure spectral element context and finally to [1] and [14] where applications of this idea are used in the context of finite element simulations

(*) Manuscript received, Februaiy 9» 1996

(*) Mathématiques pour l'Industrie et la Physique, Umte Mixte de Recherche CNRS LPS Toulouse 1 INSA Toulouse (UMR 5460), Université Paul Sabatier 118 loute de Narbonne, 11062 Toulouse Cedex 4, France

(**) Laboratoire d'Analyse Numeiique, Univeisite Pieire et Marie Cune, 75252 Pans Cedex 05, France et ONERA Parallel Computing Division

M2 AN Modélisation mathématique et Analyse numérique 0764 58^X/97/02/$ 7 00 Mathematical Modellmg and Nurnencal Analysis (Q AFCET Gauthiei Villais

290 F BEN BELGACEM Y MADAY

Applications of the mortar element method are also present in situations where domain décomposition is not (only) involved for parallel purpose but has been used to mesh the global domain Indeed, a complex geometry can often be decomposed into nonoverlapping subgeometnes that are more easily meshed mdependantly The framework of conforming approximation is then very stringent to allow for the more flexible use of this concept Indeed ït forces the interface of the subdomains to have coïncident meshes This prevents in particular, the use of tetrahedra m one subdomam with hexahedra m an adjacent one Our goal m this paper is to present and analyse the mortar element method in this particular context

2 DEFINITION OF THE METHOD

2.1. Présentation of the discrete space

The method we propose hère is adapted to the discretization of three dimensional, second order problems that are wntten under a vanational formulation in a domain Q of M3 The mam concern is then to provide an approximation of the space Hl(Q), hence, there is not much restriction m focussmg on the problem of the Laplace équation with homogeneous boundary conditions Find u e H]Q(Q) such that

| (2

(the symbol ( . , . ) is the duality painng between Hl0( Q ) and ( H l(Q)) The startmg pomt of this method is a décomposition of the domain Q where the partial differential équation is to be solved We consider a partition of Q

K

Q=\J Q^k with Q^k n Q^e = 0, if * =* 2

k= 1

With each subdomam Q^k we then associate a regular triangulation made of éléments that are either hexahedra or tedrahedra In order to avoid the techniques required for the treatment of the curved boundanes, we assume, agam for the sake of simplicity, that each subdomam Q (and thus Q ) is a polyhedron and also that each face of Q^k that meets the boundary dQ is entirely inside öQ We dénote by V* ', 1 ^ i ^ F(k) the faces of Qk Each such face mhents a triangulation made of either tnangular or quadrilatéral éléments (that are all entire faces of an element of the triangulation of Q^k) We shall assume, m what follows that these (2D) triangulations are M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis

THE MORTAR ELEMENT METHOD 291

uniformely regular. Note that, since the triangulations on two adjacent sub- domains are independant, the interface yk* = Qk n Q is provided with two different and independant (2D) meshes.

The triangulation ^Fh{Qk) being chosen over each Qk, then comes the définition of the finite element functions. We choose locally the finite element method that is best suited to the local properties of the solution. Let us assume that we work with the simple generic case of linear finite éléments. We first deflne the finite element functions locally and introducé the space

X^k(Q^k)={vle *°(fl^fc),f>J|a^O = 0, V t e

where £P\t) is the set of all linear function over t. The global finite element approximation will consist of functions whose restriction over each Qk belongs to X^h{Q^k). Since the interface is provided with two independant meshes, the contraint of continuity of the global function over Q is not compatible with good approximation properties of the discrete space. In the gênerai case, such a continuity requirement would "block" all degrees of freedom over y^{k v} Inversely, imposing no condition on the jump across the interface is also known to be a bad choice. In what follows we express the matching that is sufficient to ensure the optimality of the global approximation.

The mortar element method first deals with the skeleton of the décomposition, Le. the union of all interfaces

kj k=\

and consists in chosing one of the décompositions of Sf. This décomposition is made of mortar s, noted yw, 1 ^ m ^ M that are disjoint i.e.

M

and, in addition, satisfy the fundamental hypothesis that each mortar coincides with an en tire face of one of the subdomains, i.e.

Vm, 1 *£ m ^ M, 3fc, 1 ^ k *£ K, 3it 1 ^ i ^ F(k), ym = ƒ * ' , if a mortar coincides with both (entire) faces of connected subdomains, we choose one of the two so that there exists an application from the set {l, ..., M} into the set {l, ..., K) that associâtes to each mortar index m the corresponding subdomain index k(m). Note that each mortar is consequently meshed with (2D) éléments that are all entire faces of (3D) éléments of Qk.

292 F BEN BELGACEM Y MADAY

It is well known that the trace of the solution u of our problem over £f is of prime importance in the domain décomposition framework Indeed, would it be known, then the solution u could be computed locally within each subdomam by solvmg K independant Dinchlet problems over each element Qk This is at the basis of the Schur complement method We thus introducé a space W^h(Sf) of discretization for this trace (called the mortar space hereafter)

K,3w^kh e X^h(Q^k),

This set being defined, we introducé the space of approximation over Q

Xh=

l^k^K, Vi, 1 s= i =S F(k), ' i s a m o r t a r , ( u * ) | ^ = Ç » | / *

j

where the finite element set wkh ' is defined locally on each 1* l (actually, only on those faces that are not mortars) Let wkh l be the space of all restrictions to r * ' of the éléments of Xh In our applications, wkh ' is an appropriate subspace of wkh \ with same dimension as wj ' n H^I* l), that will be defined in the otherwise, Vy, e wkh ', j [(»J)|,* - p|/* ] V = O j (2 2)

h

next subsection

Note that this définition of the finite element space of approximation leads to a non unique définition of the values of the discrete functions on the edges of the skeleton even if the global mesh would allow for a Standard conforming définition of this space In the context of the parallel implementation, this leads to a major improvement of this formulation of the mortar element method over the previous ones as is explamed in [5] and [6] Indeed, it allows for reducing the amount of communications between different subdomams (see also [13]

for similar ideas)

The last point that needs to be adressed now is the définition of the space w^kh ' In this paper, it will be solely given in the case where the face I* ' is M2 AN Modélisation mathématique et Analyse numenque Mathematical Modelling and Numencal Analysis

THE MORTAR ELEMENT METHOD 2 9 3

meshed with triangular éléments. The case of quadrilatéral éléments is ex- plained in [4] and [7] and the gênerai case where both quadrilatéral and triangular éléments are involved is easily deduced from these two papers.

2.2. Définition of the space wh' '

We are in the case where the face J*'' is meshed with triangular éléments.

We dénote by a^/J(, 1 ^ p =S P(k, / ) , the set of all vertices of the triangles and distinguish the internai nodes that belong to ƒ*'' (numbered from 1 to P0(k,i)) from those that belong to the boundary of ƒ * ' (numbered from P⁰(k,i)+\ to P(k,i)). With all these nodes are associated the shape functions h so that any element x of vt>*'l can be written as

P(/c,0

and those éléments that belong to HlQ( T^1 f ) can be written as

The vertices a , P0(k, i) + 1 ^ p ^ P(k, i) belong to the same triangles as internai nodes within T"*1'. We dénote by a^s 1 ^ ^ ^ ö ( p ) those vertices inside f*' ' that belong to a side of a triangle with end point a . For each such /?, P⁰(/c, z ) + l ^p^P{k,i), we choose Q(p) positive real numbers À'' with

The définition of the space wkh l is then

294 F BEN BELGACEM, Y MADAY ît can also be wntten as

2.3. Présentation of the discrete problem

From the variation al formulation of the problem (2 1) together with the définition of the discrete space X^h, ît is an easy matter to defme a discrete problem correspondmg to a Galerkm approximation It consists in flndmg a solution uh G Xh such that

h

, f Vu

Vv

d

=\ p

dx

The well posedness of this problem is easy to prove It is a standard conséquence of the Pomcaré înequahty m the case where each subdomam Q has an edge on 3Q, the gênerai case is treated in [9] Thus there exists a unique solution to this problem

Another formulation of the method can also be given by making use of a Lagrangian and expressing the gluing process as an external constramt and not as a part of the définition of the discrete space This is the route followed by [14], [15] for example, see also [5]

The unknowns of the problem are then

(a) all the values of uh at the internai nodes withm each subdomam, (b) the mortar values within each mortar,

(c) the values of uh on each node of the edges of Qk

(remind that the discrete functions are multivalued on each edge) Note that the nodal values of ukh at the mterior of each face that is not a mortar are denved from the nodal values of the mortar function together with the nodal values on each edge of the correspondmg face For more details on the matrix form of the method, we refer to [6]

3. NUMERICAL ANALYSIS OF THE METHOD

This analysis follows the same hnes as the original proofs m [8] and [9] The second Strang lemma in the nonconforming situation where X^h <Z H^]Q(Ü) gives the error estimate

M2 AN Modélisation mathématique et Analyse numeiique Mathematical Modelling and Numencal Analysis

THE MORTAR ELEMENT METHOD 295

- v.

h

sup —p . (3.1)

where [vh] dénotes the jump of vh across the interfaces of £? and — dénotes the normal derivative of u. The first error term in (3.1) is known as the approximation error, the second term is the consistency error and is a consé- quence of the discontinuity of the éléments of Xh through the interface.

3.1. Analysis of the consistency error

A bound for the consistency error is derived as follows

where we remind that y. g = Qk n Qe and <p is the mortar function associated

k 2 k f

with vr Since 0 coïncides with either v or v over y ' — say here with v -— we have

y

2

' not a mortar

f M ( „* _

Jr*'dn

Usvng now (2.2), we deduce that

-k,i

" ' not a mortar

so that

du inf

1 ' not a mortar

not a mortar

[ II Ü*II » • « ( / * • ) + II011 vol 31, n° 2, 1997

296 F. BEN BELGACEM, Y MADAY

Recalling once more that (p coincides with one of the traces of vh over

deduce from the standard trace theorem that , we

not a mortar

It is well known that, for any g e H^iT*'1)

inf^ || g — W\\L2(r* ') ^ c^ II 9 I I H \ I ^ ')

so that, denoting by ph the projection operator from L2( J^1 ' ) onto w^ ', and using a interpolation argument, we have

.*-•" i 1 / 2 I j 11 ,

Besides a standard Aubin Nitsche argument leads to

\\9 -Pnign^n^.^^ ch^ïn \\g - Combining these two results yields

II d n

the trace theorem gives then

J f e = 1

From the définition of the matching, the consistency error is then of optimal order.

3.2. Analysis of the best approximation error

Let us turn now to the best approximation error. The key point of its analysis is given by the stability property of the foliowing operator defined from

2 l *^{y p} 2 *

L2( ƒ * ' ) into wkh ' n Hl0( ƒ * ' ) as follows : V* G L2( ƒ* ' ),

f

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis

THE MORTAR ELEMENT METHOD 2 9 7

L E M M A 3 . 1 : There e x i s t s a c o n s t a n t c > 0 s u c h t h a t : V / G ! ? ( ! * * ^l)^y

Before proving this result, we give, as a corollary, our main resuit con- cerning the approximation error.

THEOREM 3.2 : For any real numbers a^k, 1 ^ o^k ^ 2, there exists a constant c, independant of h such that, for any function v e H^ö( Q ) wit h

k

Proof : It follows by using the same lines as in [9]. With each function fullfilling the previous hypothesis we associate a discrete function, defined locally within each subdomain, that satisfy

(take for instance the discrete interpolate of v ). Such a function does not satisfy the matching condition across the interfaces. To cope with this, we first define the mortar function that will dérive the value of the discrete function in Xh that will approximate v. This mortar function 0, over each ym will be chosen as the restriction of v ^ ( « o . By construction, this element belongs to W^h(Sf). We shall now modify the values of v ^h over each face that is not a mortar. Let i** be such a face and define the element of w^¹ n H⁰(j*¹) by

^ ' [ ( ^ h\ae ~ (fi\re ']* T m s ele m e n t c a n ^e extended into an element of Xh(Q*), noted rh' = ^ ' ' ( ^ ' ' [ ( S h\Q* - <f>)\r* >~\ ) t n a t vanishes over each face of Q except i*\ Such a lifting operator exists as is proven in e.g. [11].

In addition, this operator can be chosen to satisfy the stability property

The new approximation of v that belongs to Xh by construction, is thus

' not a mortar

298 F BEN BELGACEM, Y MADAY

Making use of an inverse inequality over weh'1 (remind the uniform assumption done on the triangulation of each face), we dérive, in as standard manner that

recalling that (p is the trace over the mortars of v /ï, it is straightforward that

K

k= \

( K K \

\k=\ k=\ )

from (3.4) together with a trace bound, we dérive

K K

k=\ k=\

We are left now with the proof of the stability property (3.3)

3.3. Stability of the operator n^l

The proof of Lemma 3.1 is performed in two steps. For the sake of simplification, we shall skip any référence to the exponent ks l in the different notations, assuming that we are working on a référence domain e.g.

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis

THE MORTAR ELEMENT METHOD 2 9 9

The first step involves the L²( 7")-projection operator p^h over w^h deflned by :

^ e L2(F), phx e wh is such that :

The point that is adressed in this step consists in the proof of

LEMMA 3.3 : The re exists a constant a, 0 < a < 1 such that :

Proof : Let us first recall the following two formula valid on any triangle with vertex A, B, C : \/<p e

(3.6)

ï y ^

(C)) (3.7)

I2TI

+ <p(A)<p(B) + <p(B) <p(C) + <p(A)<p(C))

|2T|

i

K C)]

(3.8) Only for clarity, we shall assume that the mesh over F is structured and composed of isosceles rectangle triangles so that any node a^; on the edges of F that is not a vertex satisfies Q(p) = 2 as follows.

The proof is based on the following simple equality : V / e L²(F),

\\nhX-Phx\\L\n = J n f A IInk X - Xh\\ L\n »

and the construction of a suitable xh- Let us define ^ e w^ such that

hence

300 F BEN BELGACEM, Y MADAY

Example of the triangulation of a face.

Usmg (3 6) to compute the L -norms of this expression in the five triangles with a star, we arrive at the

nhX - Xh\

12T |

I sr I

3L^-(X„(a>2) + *»(aI3) ) +positive term , where we have wntten the only expressions where the influence of a3 will enter in Usmg the fact that xh belongs to wh, we deduce

I2TI

On the other hand, the use of (3 7) or (3.8) on each of the triangles sharing the vertex a3 allows for proving that

I2TI

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis

THE MORTAR ELEMENT METHOD 301 which proves that for the contributions where this corner arises the inequality of the Lemma holds with the constant a = j . For the contributions of the corners of F, the same strategy can be used which ends the proof.

The second step is shorter and will end the proof of Lemma 3.1. It is readily checked that

Using now (3.2) and (3.5), we dérive

(^Jf)2= (xhX-PhX)2jr\ PhXnhX>

so that using the fact that ph is a contraction in L2( F) we deduce

(^hxf^l (n

x -P

xf + \\x\\

\

11***11 LHn*

J r J r

and from lemma 3.2 we then deduc that

Kzllr-(r) ^jzr^ \\x\\

Hn-

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[1] Y. ACHDOU and O. PlRONNEAU, 1995, A fast solver for Navier-Stokes équations in the laminar régime using mortar finite élément and boundary élément methods SIAM J. Num. Anal, vol. 32, pp. 985-1016.

[2] G. ANAGNOUSTOU, 1991, Non conforming sliding spectral élément methods for the unsteady incompressible Navier-Stokes équations, PhD Thesis, Massachusetts Institute of Technology, Cambridge, Ma.

[3] G. ANAGNOUSTOU» Y. MADAY, C. MAVRIPUS and A. T. PATERA, 1990, On the

mortar élément method : generalization and implementation, Proceedings of the third International Conférence on Domain Décomposition Methods for RD.E. eds T. F. Chan, R. Glowinski, L Pénaux and O. B. Widlund, SIAM, Philadelphia, pp. 157-173.

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[5] F. BEN BELGACEM, 1994, The mortar finite élément method with Lagrange multipliers, (Rapport interne MIP, Université Paul Sabatier) (to appear).

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[6] F. BEN BELGACEM and Y. MADAY, 1994, Non conforming spectral élément methodology tuned to parallel implementation, Comp. Meth. in Applied Mech.

Eng, vol. 116, pp. 59-67.

[7] F. B E N BELGACEM and Y. MADAY, 1993, Non-conforming spectral method for second order elliptic problems in 3D, Est-West J. of Num. Math., vol. 1-4, :-:> pp. 235-252.

[8] C BERNARDI, N. DÉBÏT and Y. MADAY, 1990, Coupling spectral and finite élément methods for the Laplace équation, Math. Comput,, vol. 54-189, pp. 21- 4 1 .

[9] C. BERNARDI, Y. M A D A Y and A. T. PATERA, 1994, A new nonconforming

approach to domain décomposition : the mortar élément method, Nonlinear Partial Differential Equations and Their Applications, eds H. Brezis and J. L. Lions Pitman, New York, pp. 13-51.

[10] C. BERNARDI, Y. M A D A Y and A. T. PATERA, 1993, Domain décomposition by the mortar élément method, Asymptotic and numerical methods for partial differential équations with critical parameters, eds H. Kaper and M. Garbey, Nato ASI séries.

[11] P. E. BJORSTAD and O. B. WlDLUND, 1986, Itérative methods for the solution of elliptic problems in régions partitionned in substructures, SJAM J. Num. Anal, vol. 23, pp. 1097-1120.

[12] N. DÉBIT, 1992, La méthode des éléments avec joints dans le cas du couplage des méthodes spectrales et des éléments finis, PhD Thesis, Université Pierre et Marie Curie, Paris, France.

[13] C. FARHAT and F. X. Roux, 1991, A method of finite élément tearing and interconnecting and its parallel solution algorithm, Int. 7. Num. Meth. Engr., vol. 32, pp. 1205-1227.

[14] P. L E TALLEC and T. SASSÏ, 1995, "Domain Décomposition with Nonmatching Grids : Augmented Lagrangian Approach", Math, of Comp., vol. 64, pp. 1367-1396.

[15] P. L E TALLEC and S. RODRIGUES, 1993, Domain décomposition method with nonmatching grids applied to fluid dynamics, Finite élément in fluids, new trends and applications, eds K. Morgan, E. Onate, J. Périaux, J. Peraire and O. C. Zien- kiewics, Pineridge Press, Barcelone, pp. 418-426.

[16] C. MAVRIPLIS, 1989, Nonconforming discretization and a posteriori error esti- mations for adaptive spectral élément techniques, PhD Thesis, Massachusetts Institute of Technology, Cambridge, Ma.

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